Editing Lara documents/Lara 001.002

{{Apocrypha}}
''Translated by tobyas''

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Cahy'leh 

Your letter intrigued me to do some research. I talked to members of several guilds to try to determine what people thought of the idea. Apparently this is known fairly well and appears in many Guild training courses. It is fairly well accepted that, although we cannot know these numbers exactly, we do not really need to. However, there are more of these numbers than just "square side" 2. A member of the Guild of Stone Masons proposed this problem. I take a cube of stone and measure its volume to be 2 units by sinking it in water. What is the length of the side? Something that, when multiplied by itself twice, equals 2? A "cube side" 2. These strange numbers are simply that: strange. There are as many of them as there are branches on the Tree. (Translator Note: I believe he is talking about the Great Tree of Possibility which is a concept based around the writing of Ages.) Normally I would be tempted to agree with most others and say that it is an error in measurement. There are, after all, limits to how well we can measure. However, we are assuming conditions to the 25. Generally I was told that people tend to use estimations when building something, but when planning it out it is almost as if they are using several different types of measurements: one for 1 span, one for "square side" 2, one for "square side" 3, and so on.

Now, I know you as one who likes puzzles, so you might be interested in a puzzle proposed to me by a member of the [[Guild of Mechanists]] named Koh'shee. Start with a circle with a "dividing line" of length 2 [[span]]tee. Draw a second "dividing line" 1,000 (1,000 D'ni = 15,625 Decimal) [[toran]] different from the first. You will now have a picture of 1 / 4 of a circle. Connecting the two points on the circle and using what you discovered about the 1,000 Triangle will give you "square side" 2. Notice that this is the same as using the "horizon" on this angle. Also notice that the arc of this circle is more than "square side" 2. Now, cut the 1,000 angle into two 500 angles and make a third point on the circle. Use the “horizon”again on these two angles and add the results. Since each "horizon" is a little more than 3 / 4, the sum is a little more than 3 / 4 * 2. Keep repeating this process and you get closer and closer to the length of the arc in the circle. This value, Koh'shee tells me, is something that the Mechanists use a lot in their work with gears despite the fact that it, like "square side" 2, must be estimated. He also tells me they often estimate this when doing small projects by using 11 / 7.

From an Age writing perspective, I can tell you that these same concepts will work in every Age on the Great Tree that we deem stable and most that we deem unstable. I imagine that it would be very difficult to write an Age where these numbers did not exist as they are a construct of your mind and not of a writer. So either the whole Tree is unstable or this is not a contradiction. Either way, I wouldn’t worry too much about it. D'ni is stable enough and obviously passed the rigorous battery of safety inspections we put it through. We do live here and have been living here for our entire lives. I am absolutely convinced that the Age of D'ni is stable. Also keep in mind that the smaller the inconsistency, the longer the lifespan of the age. When a writer mixes up words and creates a big contradiction, Ages tear themselves apart as the seams rather quickly. If these numbers are an inconsistency, they are a minute one. D'ni will be around for as long as the Great Tree of Possibility.

Keep exploring but always do your research, 

Ardis

== Translator's notes ==
This document contains a lot of mathematical notation and seems to provide a lot of insight as to how the D'ni wrote and thought about mathematics. This might be a veritable Rosetta Stone of information. I have put all of the things I believe to be mathematical terms in quotes in the main body of the letter. I will now go through them and explain how I came to discern their meaning and translation. –tobyas

"Square side"—This appears to be the D'ni form of the square root. Not surprisingly, the D'ni seem to have kept their tie between geometry/design/engineering and the abstract mathematical concept of numbers by permanently paying homage to the relation between the radical and the side of the square. 

"Cube side"—Unsurprisingly, the D'ni have extended the square root to the cube root the same way we do. Unfortunately this document makes no mention of the existence of a forth or higher root. I highly doubt they would use a term like "hyper cube side." This is something to look out for in future documents. 

"Dividing line" or the alternate translation "Separating line"—used in this context it appears to directly refer to the diameter of a circle. However, I might speculate that these terms may refer to any line of symmetry in a geometric figure. 

"Horizon"—This appears to be a trigonometric function as the author makes numerous references to using it on an angle at the center of a circle to determine the distance between two points on the circle. With the approximate values given the horizon of an angle looks to be 2 * sine(angle / 2). Technically with this we could create an additional trigonometric function "cohorizon", but it is unclear if the D'ni used anything like this. An additional note about this is in regards to the name. The name appears to come from the idea that a surveyor’s view of a new age seems to be a circle. Then, the distance between two points on the horizon can be calculated knowing the angle between the two items and the visibility of the day. 

The number that would be calculated using Koh'shee's method is pi/2. This is an interesting curiosity about D'ni mathematics as this seems to be another aspect which seems based off of a quarter of a circle. On the surface, we use a measure of angles called radians. Pi/2 happens to correspond to a quarter of a circle. 

"Multiplication"—I have used the symbol * in place of the D'ni symbol of multiplication. After a little bit of research the D'ni symbol for multiplication is the letters of the word for "with" or "by" (''t''' in English letters) written on top of each other inside of a box. 

"Division"—Just like with multiplication, division is denoted by two letters on top of each other drawn inside of a box. The letters are ''te'' which corresponds to "of" as in "a member of." This is hardly surprising considering the words that Cahy'leh used in the first letter of this series.

As for the equation itself, the D'ni numerical system is rather unique in that all the numbers have boxes around them. As a result when writing a number with multiple digits the whole number is written inside of a long rectangle. This has been seen in many places all around D'ni and their ages. Here it seems that D'ni extended this to their equations. The same long rectangles are written around whole expressions seeming to indicate, truthfully, that an expression is actually just a different way to write a number. Incidentally, this may explain why we have never see equations anywhere as the D'ni simply treated them as numbers and found little use for the equations themselves in formal letters. Additionally the D'ni have a unique way of writing an equation. The advantage of their system is that it allows us to write expressions without the use of order of operations or parenthesis. So, an expression like 2 * (3 + 4) would be written ''2 34 + *'' all in a large rectangle with double lines separating the different numbers. At the very least, this particular D'ni man had this habit. As we discussed, it is possible that not all of D'ni shared this notation.

—tobyas